---
title: "Short note: Nyquist theorem"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:short-note-nyquist-theorem"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/short-note-nyquist-theorem
site: BCT Engineering
---

# Short note: Nyquist theorem

Signals and Systems · Chapter 4: Sampling

- Asked in: 2073 Bhadra, 3 marks
- Video: [Nyquist Rate & Nyquist Interval](https://www.youtube.com/watch?v=clDgsheEb9g) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/short-note-nyquist-theorem
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

Write a short note on the Nyquist theorem.

## Short answer

The Nyquist theorem states that a band-limited signal with highest frequency f_M can be exactly reconstructed from its samples if the sampling rate f_s > 2f_M. If f_s < 2f_M, spectral copies overlap and aliasing occurs. Reconstruction uses an ideal low-pass filter. In practice, an anti-aliasing filter and oversampling are used.

## Answer

The **Nyquist (sampling) theorem** states that a continuous-time signal band-limited to a highest frequency $f_M$ can be exactly recovered from its samples if it is sampled at a rate
$$
f_s > 2 f_M
$$
- $2f_M$ is the **Nyquist rate**; $T_N = 1/(2f_M)$ is the **Nyquist interval**.
- In the frequency domain, sampling repeats the spectrum $X(j\omega)$ at every multiple of $\omega_s$. With $\omega_s > 2\omega_M$ the copies do not overlap.
- Recovery is done with an ideal low-pass filter (gain $T$, cutoff between $\omega_M$ and $\omega_s - \omega_M$).
- If $f_s < 2f_M$, copies overlap and **aliasing** occurs; high frequencies appear as low ones and the signal is lost.
- In practice an anti-aliasing low-pass filter is placed before the sampler and $f_s$ is kept above $2f_M$ (e.g. audio with $f_M = 20$ kHz is sampled at 44.1 kHz).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Nyquist theorem requires
   A. fs < 2fM
   B. fs > 2fM
   C. fs = fM
   D. fs > fM
   Answer: B. Then exact recovery is possible. (Key point: Theorem)

2. Sampling at ωs causes
   A. The spectrum to compress
   B. The spectrum to vanish
   C. The spectrum to repeat at every multiple of ωs
   D. Only a shift by ωs
   Answer: C. Copies do not overlap when ωs > 2ωM. (Key point: Frequency domain)

3. If fs < 2fM
   A. Signal is recovered by an ideal filter
   B. The Nyquist interval becomes zero
   C. Only DC is affected
   D. Copies overlap and high frequencies appear as low ones
   Answer: D. Aliasing cannot be undone. (Key point: Aliasing)

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